Developmental differences in learningfrom large but rare losses

This notebook contains exploratory analyses of behavioral data collected to investigate the relationship between risk taking behavior and probabilistic learning.

The sample consists of three age groups: kids, teens and adults and we hypothesize that sensitivity to learn from high variance feedback improves with age (and this is related to better risky decisions).

Subjects completed a probabilistic learning task in the scanner, a risky decision making task (BART) outside the scanner and numerous questionnaires. The focus of this notebook is on the first task.

The plan of analysis is to establish that adults are more sensitive to high variance feedback in the probabilistic learning task and relate this (modeled) sensitivity to behavior both in BART and other self-reported risky behaviors. Details of correlations are found here

Sample info

First let’s get a sense of the sample. Here is how many subjects we have who have complete datasets for the probabilistic learning task and their age break downs.

machine_game_data_clean %>% 
  group_by(age_group) %>%
  summarise(min_age = min(calc_age),
            mean_age = mean(calc_age),
            sd_age = sd(calc_age),
            max_age = max(calc_age),
            n = n()/180)

Performance in RL task

This task is a modified Iowa Gambling Task. Subjects are presented with a fractal in each trial. The fractals represent different machines (single-armed bandits). Subjects choose to play or pass in each trial. Each machine yields a probabilistic reward. There are four machines in total. Two with positive and two with negative expected value. One of each of these machines has a low variance reward schedule while the other has a high variance reward schedule.

• One machine gives $5 90% of the time and -$495 %10 of the time
  
• One machine gives -$5 90% of the time and $495 %10 of the time
  
• One machine gives $10 50% of the time and -$100 %50 of the time
  
• One machine gives -$10 50% of the time and $100 %50 of the time

Points earned

Performance in this task can be assessed by looking at the total number of points subjects make at the end of task. The following graph shows that adults collect more points in this task compared to kids.

machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  summarise(total_points = sum(Points_earned)) %>%
  do(assign.age.info(.)) %>%
  group_by(age_group) %>%
  summarise(mean_points = mean(total_points),
            sem_points = sem(total_points)) %>%
  ggplot(aes(age_group, mean_points))+
  geom_bar(stat='identity', position = position_dodge((0.9)))+
  geom_errorbar(aes(ymin=mean_points-sem_points, ymax=mean_points+sem_points), position = position_dodge(0.9), width=0.25)+
  theme_bw()+
  xlab('Machine')+
  ylab('Mean points')+
  labs(fill='Age group')

This difference is statistically significant: adults earn more points compared to the kids.

tmp = machine_game_data_clean %>%
  group_by(Sub_id) %>%
  summarise(total_points = sum(Points_earned)) %>%
  do(assign.age.info(.))

summary(lm(total_points~age_group, data=tmp))

Call:
lm(formula = total_points ~ age_group, data = tmp)

Residuals:
    Min      1Q  Median      3Q     Max 
-2826.4  -975.1  -220.0  1134.7  2416.5 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)   
(Intercept)       277.6      249.8   1.111  0.27029   
age_groupteen     525.9      411.0   1.280  0.20480   
age_groupadult    978.8      356.5   2.746  0.00764 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1345 on 71 degrees of freedom
Multiple R-squared:  0.09618,   Adjusted R-squared:  0.07072 
F-statistic: 3.778 on 2 and 71 DF,  p-value: 0.0276

Since we are interested in the age differences between sensitivity to different feedback schedules, we should show that this difference in performance exists especially for the high variance feedback condition(s). Here is the plot of performance (total points earned) broken down by conditions.

machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  summarise(total_points = sum(Points_earned)) %>%
  do(assign.age.info(.)) %>%
  group_by(age_group, facet_labels) %>%
  summarise(mean_points = mean(total_points),
            sem_points = sem(total_points)) %>%
  ggplot(aes(facet_labels, mean_points, fill=age_group))+
  geom_bar(stat='identity', position = position_dodge((0.9)))+
  geom_errorbar(aes(ymin=mean_points-sem_points, ymax=mean_points+sem_points), position = position_dodge(0.9), width=0.25)+
  # theme_bw()+
  xlab('Machine')+
  ylab('Mean points')+
  labs(fill='Age group')

ggsave("Points_earned.jpeg", device = "jpeg", path = fig_path, width = 7, height = 5, units = "in", dpi = 450)

Running separate models for positive and negative EV machines for ease of interpretation.

tmp <- machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  summarise(total_points = sum(Points_earned)) %>%
  do(assign.age.info(.))

In the positive EV machines there is a main effect for the high variance machine. Subjects earn fewer points in the high variance condition compared to the low variance condition. There are no age differences.

summary(lm(total_points ~ age_group*facet_labels, data = tmp %>% filter(facet_labels %in% c("-10,+100", "-5,+495"))))

Call:
lm(formula = total_points ~ age_group * facet_labels, data = tmp %>% 
    filter(facet_labels %in% c("-10,+100", "-5,+495")))

Residuals:
     Min       1Q   Median       3Q      Max 
-1320.00  -354.12    80.89   436.25   975.00 

Coefficients:
                                   Estimate Std. Error t value Pr(>|t|)
(Intercept)                         1475.17      89.35  16.510   <2e-16
age_groupteen                        208.95     146.98   1.422   0.1573
age_groupadult                       218.04     127.49   1.710   0.0894
facet_labels-5,+495                 -293.62     126.36  -2.324   0.0216
age_groupteen:facet_labels-5,+495    -40.50     207.86  -0.195   0.8458
age_groupadult:facet_labels-5,+495   -99.59     180.29  -0.552   0.5815
                                      
(Intercept)                        ***
age_groupteen                         
age_groupadult                     .  
facet_labels-5,+495                *  
age_groupteen:facet_labels-5,+495     
age_groupadult:facet_labels-5,+495    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 481.2 on 142 degrees of freedom
Multiple R-squared:  0.1425,    Adjusted R-squared:  0.1123 
F-statistic:  4.72 on 5 and 142 DF,  p-value: 0.0005098

In the negative EV machines there is again a main effect for the high variance machine: Everyone losses fewer points in the low variance condition. There is also a main effect for adults: Adults perform better than kids for both negative EV machines.

summary(lm(total_points ~ age_group*facet_labels, data = tmp %>% filter(facet_labels %in% c("+10,-100", "+5,-495"))))

Call:
lm(formula = total_points ~ age_group * facet_labels, data = tmp %>% 
    filter(facet_labels %in% c("+10,-100", "+5,-495")))

Residuals:
     Min       1Q   Median       3Q      Max 
-1236.21  -389.66    28.79   393.93  1000.34 

Coefficients:
                                   Estimate Std. Error t value Pr(>|t|)
(Intercept)                         -953.79      89.90 -10.610  < 2e-16
age_groupteen                         59.68     147.88   0.404 0.687150
age_groupadult                       292.36     128.26   2.279 0.024133
facet_labels+5,-495                 -471.55     127.13  -3.709 0.000298
age_groupteen:facet_labels+5,-495     29.20     209.13   0.140 0.889157
age_groupadult:facet_labels+5,-495    57.62     181.39   0.318 0.751199
                                      
(Intercept)                        ***
age_groupteen                         
age_groupadult                     *  
facet_labels+5,-495                ***
age_groupteen:facet_labels+5,-495     
age_groupadult:facet_labels+5,-495    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 484.1 on 142 degrees of freedom
Multiple R-squared:  0.2382,    Adjusted R-squared:  0.2114 
F-statistic: 8.882 on 5 and 142 DF,  p-value: 2.333e-07

So the age diffence in performance is driven by difference in performance in negative EV machines. The question is what difference in behavior in these conditions is leading to this difference in performance?

To anticipate possible cognitive processes that will be parameterized in RL models differences can lie in: how quickly the groups learn the probabilities, how much weight they put on the outcomes and/or how much like an optimal agent they behave.

Proportion of playing

The first thing we can look at is how often subjects play versus pass. It’s hard to see any age differences when we just look at frequency of overall playing as below.

machine_game_data_clean %>%
  group_by(Sub_id, Response) %>%
  tally %>%
  group_by(Sub_id) %>%
  mutate(pct=(100*n)/sum(n)) %>%
  do(assign.age.info(.)) %>%
  group_by(age_group, Response) %>%
  dplyr::summarise(mean_pct = mean(pct),
            sem_pct = sem(pct)) %>%
  ggplot(aes(Response, mean_pct, fill = age_group))+
  geom_bar(stat='identity', position = position_dodge(0.9))+
  geom_errorbar(aes(ymin = mean_pct - sem_pct, ymax = mean_pct + sem_pct), position = position_dodge(width = 0.9), width=0.25)+
  theme_bw()+
  ylab('Percentage of trials')+
  labs(fill = 'Age group')

It is also not immediately apparent how to translate this to better performance/learning in this task but one way to think about it: If people learned perfectly they should play half of the time (always for the positive expected value trial and never for the negative expected value trials). The fact that all play proportions are above 50% suggests that nobody learns perfectly and that adults might be closest to it. But this is very crude and a better way to look at it would be to see

1. how this depends on the different machines and
   
2. how it changes throughout the task.

To get a better sense of overall behavior in different contingency states we break this proportion of playing down by machines.

Now we can see age differences in playing frequency in different conditions, particularly in the negative expected value machines (bottom row).

machine_game_data_clean %>%
  group_by(Sub_id, facet_labels, Response) %>%
  tally %>%
  group_by(Sub_id, facet_labels) %>%
  mutate(pct=(100*n)/sum(n)) %>%
  do(assign.age.info(.)) %>%
  group_by(age_group, facet_labels, Response) %>%
 summarise(mean_pct = mean(pct),
            sem_pct = sem(pct)) %>%
  ggplot(aes(Response, mean_pct, fill = age_group))+
  geom_bar(stat='identity', position = position_dodge(0.9))+
  geom_errorbar(aes(ymin = mean_pct - sem_pct, ymax = mean_pct + sem_pct), position = position_dodge(width = 0.9), width=0.25)+
  ylab('Percentage of trials')+
  facet_wrap(~facet_labels)+
  labs(fill = 'Age group')

ggsave("Prop_played.jpeg", device = "jpeg", path = fig_path, width = 8, height = 5, units = "in", dpi = 450)

The differences in points earned map directly on to proportion of choosing to play each machine:

• Adults play less than kids for both negative EV machines.
• Everyone plays the high var positive EV machine less than the low var positive EV machine.
• Everyone plays the low var negative EV machines less than the low var positive EV machine.

tmp <- machine_game_data_clean %>%
  group_by(Sub_id, facet_labels, Response) %>%
  tally %>%
  group_by(Sub_id, facet_labels) %>%
  mutate(pct_play=(100*n)/sum(n)) %>%
  filter(Response == 'play') %>%
  do(assign.age.info(.))

summary(lmer(pct_play ~ age_group*facet_labels + (1|Sub_id), data = tmp))

Linear mixed model fit by REML ['lmerMod']
Formula: pct_play ~ age_group * facet_labels + (1 | Sub_id)
   Data: tmp

REML criterion at convergence: 2610.1

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.63748 -0.72488  0.04741  0.75053  1.95807 

Random effects:
 Groups   Name        Variance Std.Dev.
 Sub_id   (Intercept)  51.5     7.176  
 Residual             457.4    21.386  
Number of obs: 296, groups:  Sub_id, 74

Fixed effects:
                                    Estimate Std. Error t value
(Intercept)                          74.2529     4.1889  17.726
age_groupteen                        10.1916     6.8906   1.479
age_groupadult                        9.5567     5.9767   1.599
facet_labels-5,+495                 -17.0115     5.6163  -3.029
facet_labels+10,-100                -28.2759     5.6163  -5.035
facet_labels+5,-495                 -10.4215     5.6163  -1.856
age_groupteen:facet_labels-5,+495    -0.7663     9.2385  -0.083
age_groupadult:facet_labels-5,+495   -1.7187     8.0132  -0.214
age_groupteen:facet_labels+10,-100  -13.2928     9.2385  -1.439
age_groupadult:facet_labels+10,-100 -23.5495     8.0132  -2.939
age_groupteen:facet_labels+5,-495   -14.4151     9.2385  -1.560
age_groupadult:facet_labels+5,-495  -27.5151     8.0132  -3.434

Correlation of Fixed Effects:
                  (Intr) ag_grpt ag_grpd f_-5,+ f_+10, f_+5,-
age_grouptn       -0.608                                     
age_gropdlt       -0.701  0.426                              
fct_-5,+495       -0.670  0.408   0.470                      
fc_+10,-100       -0.670  0.408   0.470   0.500              
fct_+5,-495       -0.670  0.408   0.470   0.500  0.500       
ag_grpt:_-5,+495   0.408 -0.670  -0.286  -0.608 -0.304 -0.304
ag_grpd:_-5,+495   0.470 -0.286  -0.670  -0.701 -0.350 -0.350
ag_grpt:_+10,-100  0.408 -0.670  -0.286  -0.304 -0.608 -0.304
ag_grpd:_+10,-100  0.470 -0.286  -0.670  -0.350 -0.701 -0.350
ag_grpt:_+5,-495   0.408 -0.670  -0.286  -0.304 -0.304 -0.608
ag_grpd:_+5,-495   0.470 -0.286  -0.670  -0.350 -0.350 -0.701
                  ag_grpt:_-5,+495 ag_grpd:_-5,+495 ag_grpt:_+10,-100
age_grouptn                                                          
age_gropdlt                                                          
fct_-5,+495                                                          
fc_+10,-100                                                          
fct_+5,-495                                                          
ag_grpt:_-5,+495                                                     
ag_grpd:_-5,+495   0.426                                             
ag_grpt:_+10,-100  0.500            0.213                            
ag_grpd:_+10,-100  0.213            0.500            0.426           
ag_grpt:_+5,-495   0.500            0.213            0.500           
ag_grpd:_+5,-495   0.213            0.500            0.213           
                  ag_grpd:_+10,-100 ag_grpt:_+5,-495
age_grouptn                                         
age_gropdlt                                         
fct_-5,+495                                         
fc_+10,-100                                         
fct_+5,-495                                         
ag_grpt:_-5,+495                                    
ag_grpd:_-5,+495                                    
ag_grpt:_+10,-100                                   
ag_grpd:_+10,-100                                   
ag_grpt:_+5,-495   0.213                            
ag_grpd:_+5,-495   0.500             0.426          

This is not surprising given what the number of points earned already showed. But now that we are looking at a behavioral measure instead of an outcome measure we might be able to quantify constructs of interest like sensitivity to variance or sensitivity to the expected values of the machines.

As a first step to translate raw playing behavior to learning I recoded the choices to be correct when a subject chooses to play a positive expected value machine and pass a negative expected value machine and incorrect when the reverse is true. If a subject is learning they should be learning to play the positive expected machines and to pass the others.

Learning

Recoding the behavior in this way gave a clearer picture of the age difference in learning of optimal behavior between the conditions. Specifically we can now look at how the probability of a correct choice changes for each age group in each condition across trials.

machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  mutate(rel_tm = 1:n()) %>%
  # ggplot(aes(scale(Trial_number), correct1_incorrect0))+
    ggplot(aes(rel_tm, correct1_incorrect0))+
  geom_line(aes(group = Sub_id, col= factor(age_group, levels=c('kid', 'teen', 'adult'))),stat='smooth', method = 'glm', method.args = list(family = "binomial"), se = FALSE, alpha=0.2)+
  geom_line(aes(col= factor(age_group, levels=c('kid', 'teen', 'adult'))),stat='smooth', method = 'glm', method.args = list(family = "binomial"), se = FALSE, alpha=1, size=2)+
  facet_wrap(~facet_labels)+
  theme_bw()+
  # xlab("Relative trial number")+
  xlab("Trial number")+
  scale_y_continuous(breaks=c(0,1))+
  labs(col="Age group")+
  ylab('Correct choice')+
  theme(legend.position = "bottom",
        panel.grid = element_blank())

ggsave("Learning.jpeg", device = "jpeg", path = fig_path, width = 8, height = 5, units = "in", dpi = 450)

EV vs Variance effects on learning

Effect of EV: Comparing positive EV to negative EV (the two rows) There is no real learning, significant change in behavior across time for the positive EV machines while there is for the negative EV machines.

Effect of variance: Comparing high var to low var (the two cols). Here there is an interaction: there is no effect of variance for the positive EV machines but there is an effect for the negative EV machines such that learning from high var is harder for kids for negative EV.

So the smaller the EV the more learning on average (for all age groups) unless the outcomes are too variable, in which case kids don’t learn from negative EV either

Looking at learning effects separately for each machine to avoid interpreting messy three-way interactions.

Adults are more likely to make correct decisions in low var positive EV machine.

summary(glmer(correct1_incorrect0 ~ age_group*scale(Trial_number)+(1|Sub_id), data = machine_game_data_clean %>% filter(facet_labels %in% c('-10,+100')), family=binomial))

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: correct1_incorrect0 ~ age_group * scale(Trial_number) + (1 |  
    Sub_id)
   Data: 
machine_game_data_clean %>% filter(facet_labels %in% c("-10,+100"))

     AIC      BIC   logLik deviance df.resid 
  2765.2   2808.0  -1375.6   2751.2     3323 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-6.2463  0.1222  0.2114  0.4679  1.9208 

Random effects:
 Groups Name        Variance Std.Dev.
 Sub_id (Intercept) 2.495    1.58    
Number of obs: 3330, groups:  Sub_id, 74

Fixed effects:
                                    Estimate Std. Error z value Pr(>|z|)
(Intercept)                         1.439718   0.307086   4.688 2.75e-06
age_groupteen                       0.867289   0.513973   1.687   0.0915
age_groupadult                      1.125890   0.450255   2.501   0.0124
scale(Trial_number)                 0.007483   0.069007   0.108   0.9136
age_groupteen:scale(Trial_number)  -0.015519   0.126326  -0.123   0.9022
age_groupadult:scale(Trial_number)  0.122810   0.112620   1.090   0.2755
                                      
(Intercept)                        ***
age_groupteen                      .  
age_groupadult                     *  
scale(Trial_number)                   
age_groupteen:scale(Trial_number)     
age_groupadult:scale(Trial_number)    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
             (Intr) ag_grpt ag_grpd sc(T_) ag_grpt:(T_)
age_grouptn  -0.591                                    
age_gropdlt  -0.672  0.409                             
scl(Trl_nm)   0.001  0.000   0.000                     
ag_grpt:(T_)  0.000 -0.001   0.000  -0.546             
ag_grpd:(T_)  0.000  0.001   0.014  -0.613  0.335      

The probability of making a correct response for the high var positive EV machine doesn’t change for adults or kids but increases for teens across trials.

summary(glmer(correct1_incorrect0 ~ age_group*scale(Trial_number)+(1|Sub_id), data = machine_game_data_clean %>% filter(facet_labels %in% c('-5,+495')), family=binomial))

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: correct1_incorrect0 ~ age_group * scale(Trial_number) + (1 |  
    Sub_id)
   Data: 
machine_game_data_clean %>% filter(facet_labels %in% c("-5,+495"))

     AIC      BIC   logLik deviance df.resid 
  3541.7   3584.4  -1763.8   3527.7     3323 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-4.5099 -0.7129  0.2903  0.5743  3.5970 

Random effects:
 Groups Name        Variance Std.Dev.
 Sub_id (Intercept) 2.234    1.494   
Number of obs: 3330, groups:  Sub_id, 74

Fixed effects:
                                   Estimate Std. Error z value Pr(>|z|)  
(Intercept)                         0.42654    0.28650   1.489    0.137  
age_groupteen                       0.53545    0.47253   1.133    0.257  
age_groupadult                      0.54340    0.41176   1.320    0.187  
scale(Trial_number)                 0.02284    0.06365   0.359    0.720  
age_groupteen:scale(Trial_number)   0.25735    0.10973   2.345    0.019 *
age_groupadult:scale(Trial_number)  0.02660    0.09651   0.276    0.783  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
             (Intr) ag_grpt ag_grpd sc(T_) ag_grpt:(T_)
age_grouptn  -0.606                                    
age_gropdlt  -0.695  0.422                             
scl(Trl_nm)   0.001  0.000   0.000                     
ag_grpt:(T_)  0.000  0.013   0.001  -0.580             
ag_grpd:(T_)  0.000  0.000   0.002  -0.660  0.383      

All groups show improvement across trials for the low var negative EV machine but adults learn faster than kids and teens.

summary(glmer(correct1_incorrect0 ~ age_group*scale(Trial_number)+(1|Sub_id), data = machine_game_data_clean %>% filter(facet_labels %in% c('+10,-100')), family=binomial))

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: correct1_incorrect0 ~ age_group * scale(Trial_number) + (1 |  
    Sub_id)
   Data: 
machine_game_data_clean %>% filter(facet_labels %in% c("+10,-100"))

     AIC      BIC   logLik deviance df.resid 
  3950.7   3993.5  -1968.4   3936.7     3323 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-4.1909 -0.8578  0.3636  0.7398  3.7967 

Random effects:
 Groups Name        Variance Std.Dev.
 Sub_id (Intercept) 1.066    1.033   
Number of obs: 3330, groups:  Sub_id, 74

Fixed effects:
                                   Estimate Std. Error z value Pr(>|z|)
(Intercept)                        -0.13317    0.20284  -0.656 0.511504
age_groupteen                       0.47515    0.33264   1.428 0.153175
age_groupadult                      1.00168    0.29041   3.449 0.000562
scale(Trial_number)                 0.30176    0.06299   4.791 1.66e-06
age_groupteen:scale(Trial_number)  -0.01954    0.10145  -0.193 0.847237
age_groupadult:scale(Trial_number)  0.33421    0.09422   3.547 0.000390
                                      
(Intercept)                           
age_groupteen                         
age_groupadult                     ***
scale(Trial_number)                ***
age_groupteen:scale(Trial_number)     
age_groupadult:scale(Trial_number) ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
             (Intr) ag_grpt ag_grpd sc(T_) ag_grpt:(T_)
age_grouptn  -0.610                                    
age_gropdlt  -0.699  0.427                             
scl(Trl_nm)  -0.003  0.002   0.003                     
ag_grpt:(T_)  0.001  0.005  -0.001  -0.620             
ag_grpd:(T_)  0.000  0.000   0.033  -0.667  0.414      

Kids don’t show learning across trials for the high var negative EV machine but adults and teens do.

summary(glmer(correct1_incorrect0 ~ age_group*scale(Trial_number)+(1|Sub_id), data = machine_game_data_clean%>% filter(facet_labels %in% c('+5,-495')), family=binomial))

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: correct1_incorrect0 ~ age_group * scale(Trial_number) + (1 |  
    Sub_id)
   Data: 
machine_game_data_clean %>% filter(facet_labels %in% c("+5,-495"))

     AIC      BIC   logLik deviance df.resid 
  3805.8   3848.6  -1895.9   3791.8     3323 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.5476 -0.6770 -0.3721  0.7510  3.1894 

Random effects:
 Groups Name        Variance Std.Dev.
 Sub_id (Intercept) 1.222    1.105   
Number of obs: 3330, groups:  Sub_id, 74

Fixed effects:
                                   Estimate Std. Error z value Pr(>|z|)
(Intercept)                        -0.92954    0.21726  -4.278 1.88e-05
age_groupteen                       0.38305    0.35560   1.077 0.281394
age_groupadult                      1.07377    0.30891   3.476 0.000509
scale(Trial_number)                 0.02305    0.06463   0.357 0.721311
age_groupteen:scale(Trial_number)   0.39636    0.10578   3.747 0.000179
age_groupadult:scale(Trial_number)  0.69180    0.09630   7.184 6.79e-13
                                      
(Intercept)                        ***
age_groupteen                         
age_groupadult                     ***
scale(Trial_number)                   
age_groupteen:scale(Trial_number)  ***
age_groupadult:scale(Trial_number) ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
             (Intr) ag_grpt ag_grpd sc(T_) ag_grpt:(T_)
age_grouptn  -0.611                                    
age_gropdlt  -0.703  0.429                             
scl(Trl_nm)  -0.002  0.001   0.002                     
ag_grpt:(T_)  0.000 -0.015   0.000  -0.611             
ag_grpd:(T_) -0.001  0.000   0.011  -0.671  0.412      

Model-free trait index

I tried to capture these effects in ‘individual difference’ variables by running the logistic regression separately for each subject in each condition. This wouldn’t capture anything different than the above analyses but I wanted to see if there were any subject-specific indices that could be correlated with other measues. I looked at three parameters:

• The intercept: whether they are more or less likely to choose the optimal action having seen half of the trials (p>0.5 if intercept>0 (i.e. log(0.5/0.5)))
• The slope: which direction and how fast the sigmoid moves in (for learning this must be positive and the larger it is the better the learning)
  
• The learning index: where in the task (i.e. scaled trial number) they are at 50% for each machine (switch point - I came up with this to capture change in both parameters. I’m not sure if it makes sense.) The smaller the better (the sooner they learn the better choice).

Because each model is run only on 45 trials the fits aren’t great and the parameter distributions have large variances.

get_learning_coef <- function(data){
  model = glm(correct1_incorrect0 ~ scale(Trial_number), family = binomial(link=logit), data = data)
  b0 = coef(model)[1]
  b1 = coef(model)[2]
  learnIndex = -b0/b1                   
  return(data.frame(b0, b1, learnIndex))
}


tmp = machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  do(get_learning_coef(.)) %>%
  do(assign.age.info(.)) 

(Error bars not shown because they are very large due to bad fits).
As expected the difference between kids and adults in slopes for the high variance negative EV machine is visible here too.

tmp %>%
  ungroup()%>%
  select(facet_labels, age_group, b0, b1, learnIndex) %>%
  gather(key, value, -facet_labels, -age_group) %>%
  group_by(age_group, facet_labels, key) %>%
  summarise(mv = median(value),
            sv = sem(value)) %>%
  ggplot(aes(facet_labels, mv, fill=age_group))+
  geom_bar(stat="identity", position = position_dodge())+
  # geom_errorbar(aes(ymin = mv-sv, ymax = mv+sv), position = position_dodge(width = 0.9), width=0)+
  facet_wrap(~key, scale="free")+
  theme(legend.position = "bottom",
        legend.title = element_blank())+
  xlab("")+
  ylab("Median value")

But it’s not a good idea to look for group differences in these parameters as they are highly variable due to bad fits from few trials.

Variance vs. EV sensitivity

Does it makes sense to look at these separately?

Since the machines differ in the variance of the outcomes and expected values it might seem sensible to look at which of these attributes has a larger effect on performance.

It’s tempting to tease apart the relative importance of these attributes for the high variance negative EV machine where we observe the performance difference between age groups.

BUT these attributes are correlated. So we can’t look at their effects separately in the same model.

#Function to calculate observed variance and observed expected value based on outcomes in trials that the subject has played.
get_obs_var_ev <- function(data){
  
  new_data = data
  new_data$obs_var <- NA
  new_data$obs_ev <-  NA
  
  for(i in 1:nrow(new_data)){
    if(i == 1){
      obs = 0
      obs_ev = 0
      obs_var = 0
    }
    else{
      #get all the trials until the current trial
      obs = new_data[1:i,]
      #filter only played trials; their belief should not be updated based on the trials they haven't played
      obs = obs %>% filter(Response == "play") %>% ungroup() %>% select(Points_earned)
      obs_var = var(obs)
      obs_probs =  as.numeric(prop.table(table(obs)))
      obs_rewards = as.numeric(names(prop.table(table(obs))))
      obs_ev = sum(obs_probs*obs_rewards)
    }
    new_data$obs_var[i] = obs_var
    new_data$obs_ev[i] = obs_ev
  }
  new_data$obs_var = ifelse(is.na(new_data$obs_var), 0, new_data$obs_var)
  return(new_data)
}

tmp = machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  do(get_obs_var_ev(.))

tmp %>%
  ggplot(aes(obs_var, obs_ev))+
  geom_point()+
  facet_wrap(~facet_labels, scales="free")+
  xlab("Observed variance")+
  ylab("Observed EV")

What we are interested in is the effect of beliefs about the machines on behavior. These beliefs can be summarized quantitatively in an ‘expected value.’

The cognitive processes that can differ with respect to this expected value can be how quickly it approaches the true expected value of a machine (the rate at which one incorporates each new data point to existing beliefs) and how truthfully the expected values are evaluated (is the utility of the expected value the same as its value).

These two processes can be captured as the learning rate and the exponent on the prediction error in an RL model.

Before moving on to modeling results here I plot the effect of observed EV (not model based) on choice to confirm that it makes sense and captures the behavioral effect:
The higher the EV of a machine the more likely it is to be played. This is the correct action for the positive EV machines but incorrect action for the negative EV machines. The behavioral effect in the high var negative EV machine is captured again with the diverging lines for age groups at low EVs.

tmp %>%
  ggplot(aes(obs_ev, correct1_incorrect0))+
  geom_line(aes(group = Sub_id, col= age_group),stat='smooth', method = 'glm', method.args = list(family = "binomial"), se = FALSE, alpha=0.2)+
  geom_line(aes(col= age_group),stat='smooth', method = 'glm', method.args = list(family = "binomial"), se = FALSE, alpha=1, size=2)+
  facet_wrap(~facet_labels, scales='free')+
  xlab("EV of played trials")+
  scale_y_continuous(breaks=c(0,1))+
  labs(col="Age group")+
  ylab('Correct')+
  theme(legend.position = "bottom",
        legend.title = element_blank())

Additional behavioral patterns

Though I focus on learning behavior and specifically difference in learning for the high variance negative EV machine there are other possible behavioral patterns that might also differ between the age groups. Here I list some examples.

Initial exploration

Do people ‘explore’ the first 10 trials where the reward probabilities for each machine are presented?

They explore less when they encounter a loss early on. In the high var pos EV machine they get 4 (small) losses in a row; in the low var negative EV machine they get a moderate loss in the first trial.

machine_game_data_clean %>% 
  group_by(Sub_id, facet_labels) %>%
  slice(1:10) %>%
  summarise(num_explored = sum(ifelse(Response == "play", 1,0))) %>%
  do(assign.age.info(.)) %>%
  ungroup() %>%
  group_by(age_group, facet_labels) %>%
  summarise(mean_num_explored = mean(num_explored/10*100),
            sem_num_explored = sem(num_explored/10*100)) %>%
  ggplot(aes(facet_labels, mean_num_explored, fill = age_group))+
  geom_bar(stat="identity",position = position_dodge(0.9))+
  geom_errorbar(aes(ymax = mean_num_explored+sem_num_explored, ymin = mean_num_explored-sem_num_explored), position = position_dodge(width = 0.9), width=0.25)+
  theme(legend.title = element_blank())+
  ylab("Percentage of exploration")+
  xlab("")

Memory effect

How does performance change depending on the delay between the last time a machine was played?

Can we think of this as a ‘memory effect’? The more trials since the last time you have played a machine, the more forgetting/interference?

For positive EV machines this is true for all groups. This is evident in the decreasing probability of a correct response the longer it has been since the last time a machine was played.

For negative EV machines adults and teens continue to make correct choices even if a lot of trials have passed since they last played that machine. Kids don’t seem to remember that the machine is ‘bad’ and are more likely to make an incorrect choice (and play the machine) the longer it’s been since they last played it.

machine_game_data_clean %>%
  group_by(Sub_id) %>%
  mutate(played_trial_number = ifelse(Response == "play", Trial_number, NA)) %>%
  mutate(played_trial_number = na.locf(played_trial_number, na.rm=F)) %>%
  filter(Trial_number > 1) %>%
  mutate(trials_since_last_played = Trial_number - lag(played_trial_number)) %>%
  ggplot(aes(trials_since_last_played, correct1_incorrect0, col = age_group))+
  geom_line(stat='smooth', method = 'glm', method.args = list(family = "binomial"), alpha=1, size=2)+
  facet_wrap(~facet_labels)+
  theme(legend.title = element_blank())+
  xlab("Trials since last played")+
  ylab("Correct")+
  scale_y_continuous(breaks=c(0,1))

Post-loss behavior

If subjects are sensitive to losses and learning something about the machines in a way that overweights their most recent experience with the machine one sanity check is to compare how many trials it takes subjects to play a machine again after a loss versus a gain. Presumably the former would be higher than the latter. One might hesitate to play a machine again after a loss but be more likely to play it after a gain.

count.postoutcome.trials <- function(subject_data){
  
  loss_trials = which(subject_data$Points_earned<0)
  
  gain_trials = which(subject_data$Points_earned>0)
  
  play_trials= which(subject_data$Response == "play")
  
  post_loss_trials = play_trials[which(play_trials %in% loss_trials)+1]
  
  post_gain_trials = play_trials[which(play_trials %in% gain_trials)+1]
  
  num_trials_post_loss = post_loss_trials - loss_trials
  
  num_trials_post_gain = post_gain_trials - gain_trials
  
  if(length(num_trials_post_gain)>length(num_trials_post_loss)){
    num_trials_post_loss <- c(num_trials_post_loss, rep(NA, length(num_trials_post_gain) - length(num_trials_post_loss)))
  }
  else if(length(num_trials_post_gain)<length(num_trials_post_loss)){
    num_trials_post_gain <- c(num_trials_post_gain, rep(NA, length(num_trials_post_loss) - length(num_trials_post_gain)))
  }
  
  return(data.frame(num_trials_post_loss = num_trials_post_loss, num_trials_post_gain = num_trials_post_gain))
}

The plot below shows the average number of trials it takes a subject to play a given machine after experiencing a loss or a gain.

For everyone and for every machine the average number of trials it takes a subject to play following a loss is higher than the average number of trials it take them to play following a gain. This suggests that subjects are responding to outcomes in a way overweights their most recent experience with the machine.

tmp = machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  do(count.postoutcome.trials(.))  %>%
  do(assign.age.info(.)) %>%
  ungroup() %>%
  select(facet_labels, age_group, num_trials_post_loss, num_trials_post_gain, Sub_id) %>%
  gather(key, value, -facet_labels, -age_group, -Sub_id) %>%
  mutate(key = gsub("num_trials_post_", "", key)) 

tmp %>%
  group_by(facet_labels, age_group, key) %>%
  summarise(mean_post = mean(value, na.rm=T),
            sem_post = sem(value)) %>%
  ggplot(aes(age_group, mean_post, shape=key, col=age_group))+
  geom_point(size=2)+
  geom_errorbar(aes(ymin = mean_post-sem_post, ymax = mean_post+sem_post), width=0)+
  facet_wrap(~facet_labels)+
  ylab("Number of trials until next play")+
  xlab("")+
  theme(legend.title = element_blank())+
  guides(color=FALSE)

Reflecting the global behavior in proportion of playing in each condition adults take longer to play after large losses in the high variance negative EV condition compared to kids while kids are less sensitive to the magnitude of loss.

summary(lm(value~age_group*facet_labels,tmp %>%filter(key=="loss")))

Call:
lm(formula = value ~ age_group * facet_labels, data = tmp %>% 
    filter(key == "loss"))

Residuals:
    Min      1Q  Median      3Q     Max 
-1.8387 -0.4617 -0.4160 -0.2342 26.8007 

Coefficients:
                                    Estimate Std. Error t value Pr(>|t|)
(Intercept)                          1.41929    0.07401  19.177  < 2e-16
age_groupteen                       -0.18511    0.11724  -1.579 0.114438
age_groupadult                      -0.14424    0.10301  -1.400 0.161503
facet_labels-5,+495                  0.25664    0.09751   2.632 0.008527
facet_labels+10,-100                 0.78004    0.11960   6.522 7.81e-11
facet_labels+5,-495                  0.56908    0.18936   3.005 0.002670
age_groupteen:facet_labels-5,+495   -0.02910    0.15382  -0.189 0.849940
age_groupadult:facet_labels-5,+495  -0.11572    0.13528  -0.855 0.392376
age_groupteen:facet_labels+10,-100   0.35057    0.19749   1.775 0.075961
age_groupadult:facet_labels+10,-100  0.78089    0.18237   4.282 1.90e-05
age_groupteen:facet_labels+5,-495    0.40507    0.31393   1.290 0.197020
age_groupadult:facet_labels+5,-495   0.99458    0.28832   3.450 0.000567
                                       
(Intercept)                         ***
age_groupteen                          
age_groupadult                         
facet_labels-5,+495                 ** 
facet_labels+10,-100                ***
facet_labels+5,-495                 ** 
age_groupteen:facet_labels-5,+495      
age_groupadult:facet_labels-5,+495     
age_groupteen:facet_labels+10,-100  .  
age_groupadult:facet_labels+10,-100 ***
age_groupteen:facet_labels+5,-495      
age_groupadult:facet_labels+5,-495  ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.616 on 3936 degrees of freedom
  (1653 observations deleted due to missingness)
Multiple R-squared:  0.06843,   Adjusted R-squared:  0.06582 
F-statistic: 26.28 on 11 and 3936 DF,  p-value: < 2.2e-16

What affects whether a subject plays after experiencing a loss? The magnitude of the loss they expired? How long they think before playing?

First look at how this changes across the task: Subject are more likely to be make the correct choice for the negative EV machines as the task goes by.

machine_game_data_clean %>%
  mutate(losstrial = ifelse(Points_earned<0,1,0),
         postloss = lag(losstrial),
         postloss_play1_pass0 = ifelse(postloss == 1 & Response == "play",1, ifelse(postloss==1 & Response == "pass", 0, NA))) %>%
  group_by(Sub_id, facet_labels) %>%
  mutate(rel_trial = 1:n()) %>%
  # ggplot(aes(rel_trial, postloss_play1_pass0))+
  ggplot(aes(rel_trial, correct1_incorrect0))+
  geom_smooth(aes(col=age_group), method='glm', method.args = list(family = "binomial"))+
  facet_wrap(~facet_labels)+
  scale_y_continuous(breaks=c(0,1))+
  theme(legend.title = element_blank())+
  xlab("Trial number")+
  ylab("Probability of correct following a loss")

tmp = machine_game_data_clean %>%
  mutate(losstrial = ifelse(Points_earned<0,1,0),
         postloss = lag(losstrial),
         postloss_play1_pass0 = ifelse(postloss == 1 & Response == "play",1, ifelse(postloss==1 & Response == "pass", 0, NA)),
         lastlossamt = lag(Points_earned)) %>%
  filter(postloss==1)


tmp %>%
  ggplot(aes(Reaction_time, correct1_incorrect0))+
  geom_smooth(aes(col=age_group), method='glm', method.args = list(family = "binomial"))+
  facet_wrap(~facet_labels)+
  scale_y_continuous(breaks=c(0,1))+
  theme(legend.title = element_blank())+
  xlab("RT")+
  ylab("Probability of correct following a loss")

Baseline is -10,+100. Less likely to be correct in any of the machines after a loss compared to this baseline. Adults are more likely to be correct following a loss for all machines. There is also an effect of response time. The longer a decision takes the less likely it is to be correct. This is even stronger for adults (they are usually faster than kids but when they do take long they are even less likely to be correct).

If slower decisions are more likely to be incorrect would this suggest less of a drift process but more interference/uncertainty about knowledge on that machine instead?

summary(glmer(correct1_incorrect0 ~ facet_labels+scale(Reaction_time)*age_group+(1|Sub_id), tmp, family="binomial"))

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: 
correct1_incorrect0 ~ facet_labels + scale(Reaction_time) * age_group +  
    (1 | Sub_id)
   Data: tmp

     AIC      BIC   logLik deviance df.resid 
  4317.3   4380.4  -2148.7   4297.3     4058 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-5.3908 -0.7149  0.3125  0.6121  4.9613 

Random effects:
 Groups Name        Variance Std.Dev.
 Sub_id (Intercept) 0.725    0.8514  
Number of obs: 4068, groups:  Sub_id, 74

Fixed effects:
                                    Estimate Std. Error z value Pr(>|z|)
(Intercept)                          1.47089    0.18790   7.828 4.95e-15
facet_labels-5,+495                 -0.83557    0.11790  -7.087 1.37e-12
facet_labels+10,-100                -1.48368    0.11610 -12.779  < 2e-16
facet_labels+5,-495                 -2.44639    0.12166 -20.109  < 2e-16
scale(Reaction_time)                -0.30295    0.05896  -5.139 2.77e-07
age_groupteen                        0.33118    0.27865   1.189 0.234622
age_groupadult                       0.82255    0.24460   3.363 0.000772
scale(Reaction_time):age_groupteen   0.06909    0.10298   0.671 0.502308
scale(Reaction_time):age_groupadult -0.28319    0.09359  -3.026 0.002480
                                       
(Intercept)                         ***
facet_labels-5,+495                 ***
facet_labels+10,-100                ***
facet_labels+5,-495                 ***
scale(Reaction_time)                ***
age_groupteen                          
age_groupadult                      ***
scale(Reaction_time):age_groupteen     
scale(Reaction_time):age_groupadult ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
                    (Intr) f_-5,+ f_+10, f_+5,- sc(R_) ag_grpt ag_grpd
fct_-5,+495         -0.373                                            
fc_+10,-100         -0.373  0.616                                     
fct_+5,-495         -0.360  0.598  0.621                              
scl(Rctn_t)         -0.092  0.071  0.026  0.061                       
age_grouptn         -0.543 -0.006 -0.018 -0.022  0.044                
age_gropdlt         -0.614 -0.011 -0.028 -0.045  0.049  0.424         
scl(Rctn_tm):g_grpt  0.057 -0.032 -0.027 -0.061 -0.573 -0.026  -0.027 
scl(Rctn_tm):g_grpd  0.057 -0.040 -0.032 -0.019 -0.628 -0.028  -0.056 
                    scl(Rctn_tm):g_grpt
fct_-5,+495                            
fc_+10,-100                            
fct_+5,-495                            
scl(Rctn_t)                            
age_grouptn                            
age_gropdlt                            
scl(Rctn_tm):g_grpt                    
scl(Rctn_tm):g_grpd  0.360             

Loss aversion

If one knew which were positive and which negative EV machines one would either always play for positive EV machines or never play for negative EV machines regardless of the observed outcome. So for the positive EV machines there would be no difference between gains/losses and for the negative EV machines there will be no points to plot (because it will never be played). The difference in behavior depending on the valence of the recently observed outcome (gain/loss) could be due to at least two reasons: memory or loss aversion. Or perhaps stronger memories for losses for adults. Do kids play the bad machine because they can’t remember how bad that machine is or they don’t care to loose as much? Perhaps there is something interesting to look at in the hippocampal activity following losses versus gains.

Studies that compute loss aversion present subjects with gambles where the amounts and probabilities are known. This is not the case for our paradigm (which is what makes it a learning task) which is why I estimate them as part of RL models later too. For the sake of it let’s assume subjects knew the gain and loss amounts for each machine and calculate loss aversion:

We don’t find a difference in the estimates between adults and kids. Neither had Barkley-Levenson et al. (2014).

get_loss_aversion = function(data){
  data = data %>%
  filter(Response != "time-out") %>%
  mutate(play1_pass0 = ifelse(Response=="pass", 0,1),
         gain_mag = as.numeric(gain_mag),
         loss_mag = as.numeric(loss_mag))

  m = glm(play1_pass0 ~ gain_mag+loss_mag, data, family="binomial")

  loss_ave = -coef(m)[3]/coef(m)[2]
  
  return(data.frame(loss_ave = loss_ave))
}

machine_game_data_clean %>%
  group_by(Sub_id) %>%
  do(get_loss_aversion(.)) %>%
  do(assign.age.info(.)) %>%
  ggplot(aes(log(loss_ave), fill=age_group))+
  geom_density(alpha=0.4, color=NA)+
  theme(legend.title = element_blank())

Cross-talk between machines

Are subjects less likely to play overall after a loss or only less likely to play that machine after a loss for that machine?

mean.postloss.play.prob <- function(subject_data){
  
  loss_trials = which(subject_data$Points_earned<0)
  
  mean_post_loss_prob <- mean(ifelse(subject_data$Response[loss_trials+1] == "play", 1, 0), na.rm=T)
  
  return(data.frame(mean_post_loss_prob=mean_post_loss_prob))
}

Probability of playing following a loss depends on machine type. Looking at all trials masks this difference. Subjects seem to learn machine specifically and cross-talk isn’t evident here.

tmp = machine_game_data_clean %>%
  group_by(Sub_id) %>%
  do(mean.postloss.play.prob(.)) %>%
  mutate(facet_labels = "all_trials")

machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  do(mean.postloss.play.prob(.)) %>%
  rbind(tmp) %>%
  do(assign.age.info(.)) %>%
  group_by(age_group, facet_labels) %>%
  summarise(mp = mean(mean_post_loss_prob,na.rm=T),
            sp = sem(mean_post_loss_prob)) %>%
  ggplot(aes(facet_labels, mp, fill=age_group))+
  geom_bar(stat="identity",position=position_dodge())+
  geom_errorbar(width=0, aes(ymin = mp-sp, ymax = mp+sp), position = position_dodge(width=0.9))+
  xlab("")+
  ylab("Post loss play probability")+
  theme(legend.title = element_blank())

Response time differences

machine_game_data_clean %>%
  ggplot(aes(log(Reaction_time))) +
  geom_density(aes(fill = age_group), alpha=0.5, color=NA) +
  facet_wrap(~facet_labels)+
  theme(legend.title = element_blank())+
  ylab("")+
  xlab("Log Response Time")

machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  summarise(mean_log_rt = mean(log(Reaction_time)),
            sem_log_rt = sem(log(Reaction_time))) %>%
  do(assign.age.info(.)) %>%
  ggplot(aes(age_group, mean_log_rt))+
  geom_boxplot(aes(fill=age_group))+
  facet_wrap(~facet_labels)+
  theme(legend.position = "none")+
  ylab("Mean Log Rt")+
  xlab("Age group")

Both teens and adults are faster than kids in all conditions but the high var negative EV.

#summary(lmer(log(Reaction_time) ~ age_group*facet_labels +(1|Sub_id), data = machine_game_data_clean))

summary(lmer(log(Reaction_time) ~ age_group +(1|Sub_id), data = machine_game_data_clean%>%filter(facet_labels == "-10,+100")))

Linear mixed model fit by REML ['lmerMod']
Formula: log(Reaction_time) ~ age_group + (1 | Sub_id)
   Data: machine_game_data_clean %>% filter(facet_labels == "-10,+100")

REML criterion at convergence: 3467.5

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.5493 -0.6537 -0.1098  0.5737  3.9100 

Random effects:
 Groups   Name        Variance Std.Dev.
 Sub_id   (Intercept) 0.03498  0.1870  
 Residual             0.15674  0.3959  
Number of obs: 3330, groups:  Sub_id, 74

Fixed effects:
               Estimate Std. Error t value
(Intercept)     6.99491    0.03642 192.072
age_groupteen  -0.20684    0.05991  -3.453
age_groupadult -0.22138    0.05196  -4.260

Correlation of Fixed Effects:
            (Intr) ag_grpt
age_grouptn -0.608        
age_gropdlt -0.701  0.426 

summary(lmer(log(Reaction_time) ~ age_group +(1|Sub_id), data = machine_game_data_clean%>%filter(facet_labels == "-5,+495")))

Linear mixed model fit by REML ['lmerMod']
Formula: log(Reaction_time) ~ age_group + (1 | Sub_id)
   Data: machine_game_data_clean %>% filter(facet_labels == "-5,+495")

REML criterion at convergence: 4024.7

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-6.5631 -0.6513 -0.1148  0.6295  3.1253 

Random effects:
 Groups   Name        Variance Std.Dev.
 Sub_id   (Intercept) 0.0469   0.2166  
 Residual             0.1848   0.4299  
Number of obs: 3330, groups:  Sub_id, 74

Fixed effects:
               Estimate Std. Error t value
(Intercept)     6.99842    0.04194 166.870
age_groupteen  -0.14905    0.06899  -2.161
age_groupadult -0.13063    0.05984  -2.183

Correlation of Fixed Effects:
            (Intr) ag_grpt
age_grouptn -0.608        
age_gropdlt -0.701  0.426 

summary(lmer(log(Reaction_time) ~ age_group +(1|Sub_id), data = machine_game_data_clean%>%filter(facet_labels == "+10,-100")))

Linear mixed model fit by REML ['lmerMod']
Formula: log(Reaction_time) ~ age_group + (1 | Sub_id)
   Data: machine_game_data_clean %>% filter(facet_labels == "+10,-100")

REML criterion at convergence: 3767

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.2565 -0.6809 -0.1059  0.6505  3.1132 

Random effects:
 Groups   Name        Variance Std.Dev.
 Sub_id   (Intercept) 0.02916  0.1708  
 Residual             0.17241  0.4152  
Number of obs: 3330, groups:  Sub_id, 74

Fixed effects:
               Estimate Std. Error t value
(Intercept)     7.03189    0.03373 208.479
age_groupteen  -0.12876    0.05548  -2.321
age_groupadult -0.12762    0.04812  -2.652

Correlation of Fixed Effects:
            (Intr) ag_grpt
age_grouptn -0.608        
age_gropdlt -0.701  0.426 

summary(lmer(log(Reaction_time) ~ age_group +(1|Sub_id), data = machine_game_data_clean%>%filter(facet_labels == "+5,-495")))

Linear mixed model fit by REML ['lmerMod']
Formula: log(Reaction_time) ~ age_group + (1 | Sub_id)
   Data: machine_game_data_clean %>% filter(facet_labels == "+5,-495")

REML criterion at convergence: 3803.6

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-6.0339 -0.6841 -0.0940  0.6244  3.9074 

Random effects:
 Groups   Name        Variance Std.Dev.
 Sub_id   (Intercept) 0.03425  0.1851  
 Residual             0.17381  0.4169  
Number of obs: 3330, groups:  Sub_id, 74

Fixed effects:
               Estimate Std. Error t value
(Intercept)     6.98863    0.03625 192.779
age_groupteen  -0.08277    0.05963  -1.388
age_groupadult -0.09490    0.05172  -1.835

Correlation of Fixed Effects:
            (Intr) ag_grpt
age_grouptn -0.608        
age_gropdlt -0.701  0.426 

Non-learners

How would you group learners vs. non-learners? Those who are more likely to make correct choices later in the task - so positive slope for the sigmoid?

tmp = machine_game_data_clean %>%
  group_by(Sub_id, facet_labels) %>%
  do(get_learning_coef(.)) %>%
  do(assign.age.info(.)) %>%
  mutate(learner = ifelse(b1>0,1,0))

with(tmp, table(learner, facet_labels, age_group))

, , age_group = kid

       facet_labels
learner -10,+100 -5,+495 +10,-100 +5,-495
      0       14      14       12      18
      1       15      15       17      11

, , age_group = teen

       facet_labels
learner -10,+100 -5,+495 +10,-100 +5,-495
      0       10       4        5       6
      1        7      13       12      11

, , age_group = adult

       facet_labels
learner -10,+100 -5,+495 +10,-100 +5,-495
      0       13      11        4       7
      1       15      17       24      21

non_learners = tmp %>%
  filter(facet_labels %in% c("+5,-495", "+10,-100")) %>%
  filter(learner == 0)

non_learners = unique(non_learners$Sub_id)
non_learners

 [1] 100003 100009 100042 100051 100057 100059 100063 100068 100105 100109
[11] 100129 100143 100169 100180 100185 100188 100207 100241 100243 100244
[21] 100250 200056 200085 200133 200162 200164 200168 200199 200211 306587
[31] 311047 311283 311444 311479 400742 407260 408394 411477

Or trials post-learning? [probably for imaging]

RL modeling

Details of model comparison can be found in a separate notebook.